Hirsch-sharpness conjecture for 3-way transportation polytopes

Let $P$ be a non-degenerate $3$-way transportation polytope of size $p \times q \times s$, with $p,q,s \geq 3$, defined by $1$-marginals. Let $G(P)$ be its graph, let $d=pqs-p-q-s+2$ be its dimension, and let $n \leq pqs$ be the number of facets of $P$.

Hirsch-sharpness conjecture for 3-way transportation polytopes.

$$\operatorname{diam}(G(P))=n-d.$$

The claim predicts that these transportation polytopes attain the Hirsch bound exactly. The supplied material gives no resolution status.